Part I: Complex Variables, Lecture 3: Conformal Mappings Instructor: Herbert Gross View the complete course: ocw.mit.edu/RES... License: Creative Commons BY-NC-SA More information at ocw.mit.edu/terms More courses at ocw.mit.edu
Thanks for the clarification. It has been over 40 years since I made the videos and that makes it difficult for me to remember what I included and what I didn’t. My intent was to never “pull a rabbit out of the hat”. In fact my goal was to attain total transparency. At age 84 I feel fortunate to be able to remainin cotact with my viewers
Just wanted to say, Herbert Gross will be remembered by all of us for teaching us and 1000s more a beautiful branch of mathematics. Thanks Alot Herbert Gross!
This series is absolutely priceless. This is the "why" that I was so badly missing in my notes. I also like the way each topic is clearly laid out, step by step in advance. Excellent work, truly, excellent!
Its amazing to think this was so eloquently teached by the brilliant Herb Gross almost 40 years ago and we are able to grasp the same fresh enthusiasm and lucidity as if we were watching this live. We live in an amazing time, where our digital legacy lingers on for, perhaps, forever.
Thanks, Tiago. Your message really resonates with me. In essence the Internet has cloned me. Every viewer in every location and at any point in time sees exactly the same thing. Moreover, long after this 88 year old Herb Gross is gone, the 42 year old Herb Gross in the videos will still be welcoming new viewers with a friendly “Hi”. I feel that “Calculus Revisited” is part of the legacy that I will leave behind for others to both enjoy and learn from. I have to thank MIT and the Rosenbaum Foundation for bestowing this blessing on me.
Your comment reminds me that the purpose of part 3 was to give students who had finished Parts 1 and 2 an insight to the most likely math courses they would take next. We felt that differential equations, linear algebra and complex variables would be the most likely choices. Our mini-series for each of the three courses was meant to be an overview rather than a complete course. I am pleased that some 40 years later, the course is still being used in this way by many viewers such as yourself.
That is very pleasing to me. When I made this mini-series all I hoped to accomplish was to give a potential student an overview of what a course in complex variables entailed.
Herb, I do research at a university and I also teach. I can appreciate the quality of the explanation here both from a learning and teaching perspective. This is how all maths should be - accessible, clear and easy to follow. Excellent, thank you.
+Danny Hoskin Thanks Danny. I always enjoy hearing from viewers and I am especially pleased when I get a nice message from a fellow teacher. While it is labor-intensive, there is a great advantage to making a video lecture. Morespecifically it takes me a long time to perfect what I want to write on the board but (1) once done, the written board provides a script that is easy to follow and (2) once the format is perfected the lecture has been given, it can be used and reused “forever”. With a bit of luck, the vide series will still be helping students even when the present herb Gross is no longer here.
I'm watching these videos from france. Surprisingly, I can easily understand each word enunciated by Professor Gross. The clarity of the speech matches the clarity of the thought.
I have to admit that at the very beginning the black and white video seemed to me quite daunting when it talks something relatively advanced. But later I found that Herbert Gross just out-rates many of the contemporary "colorful" lecturers! The entry point for complex analysis presented in this series cannot be more intuitive and understandable. Instead of loads of abstract ideas, everything just comes up naturally and appears to be applicable to "real" world. Big thumb to Prof. Gross!
I very much appreciate the kind comments you’ve written about the complex variables mini-course. Although I was enrolled in a math Ph.D. program at MIT my main interest was in teaching rather1 than in research. And during my 50 years as a math instructor I never felt that I worked a day in my life. In terms of sports analogy it would be that I chose a coaching career over playing career. Keep in mind that many great coaches were mediocre (or worse) players; and I feel that such an analogy applies to me. I was not an outstanding mathematician but I delighted in being able to teach others to be even better than I was. At the other end of the spectrum I am developing my own website (www.mathasasecondlanguagr.com) where I am trying to help viewers gain self-confidence when it comes to dealing with basic mathematics.
Actually the “one-take” success is the result of much preplanning. We wrote several mock-ups of the boards to make sure everything we wanted to say could be said and in doing that we reviewed the material several times. Finally with the boards prewritten they made wonderful “cue cards” for what I should say next. By the way, another advantage of prewriting the boards is that it leaves more time for presenting the material. Writing as we talk is very time consuming!
Thanks Greg. It is always a good feeling to know that my work has been helpful to a student; and it is especially nice when the compliment occurs some 40 years after the work was first produced. When we made the "Calculus Revisited" series we wanted to balance the "how" with the "why". We felt that once students understand "why" it makes it easier for them to appreciate and better comprehend the "why".
These lectures are priceless ! I have been trying to make heads and tails of complex numbers for a while and how to use them in real problems. I thought i was dumb because I couldnt understand it completely. Well, this made it crystal clear to me, thank you very much sir !!
What I find truly great about these lectures is not only are they very informative, but you also describe the significance and importance of a lot of these concepts. Too often, a lot of professors over complicate relatively simple concepts by being extremely rigorous, which is not the best approach to teach a concept to someone who hasn't seen it before.
Thank you for the kind comment, Elliot. A talk I used to give was entitled “Rigor Is A Function of the ‘Rigoree’ ”. The gist is that if students are willing to accept certain statements as being “true” or Self-evident”, we should accept these as being reasonable assumptions and then show them how not-so-obvious results can be deduced from the assumptions they are willing to accept. As the students mature mathematically we can backtrack and look more deeply into what the students might previously have thought to be obvious or simple. This is precisely the way we teach kids to play games. We start with showing them basic things and don’t go to the more advanced/subtle things until we feel that the kids are ready to internalize it. More to the point, how many young children would engage in an organized sport (such as Little League Baseball) if the prerequisite was to study and pass a test on a 200 page rule book?
by far the best lecture I have heard reg 'why' and 'how' we use conformal mapping..!!! Prof.Herb Gross..you had explained the entire concept from a beautiful stand point... truly brilliant .!!! thanks a lot...
Robert In between multiplication and DeMoivre’s Theorem I talked about the complex conjugate and as an application I showed how to divide two complex numbers. However, I did not use the division symbol but instead wrote (a + bi)/(c + di).
I am amazed that we are able to hold a discussion with the presenter of this material some 40 years after it was created, which would have been entirely inconceivable at the time of its creation. Fantastic video!
I have taken a course in complex variables before, but quite frankly it was uninteresting and filled with pages of definitions. This course however, is well delivered, quick to the point, mentions applications and most importantly is enjoyable! Thanks so much for these lectures!
Hi Robert, I may be missing the point you are making but as I see it, the procedure for dividing complex numbers was discussed in Lecture #1, using both Cartesian and polar coordinates.
How are you doing Mr Gross ? I just came back to your lectures after 3 years : they were really useful to me in my first engineering year and I was really happy to find that you had done something on this subject too. I am thankful to you and hope you are doing well.
Thanks, Professor. MIT is a top uni teaching things with clarity, letting the students to be creative afterwards. I'm analyzing Nash embedding theorem on Riemann Surfaces via Analytic Functions.
I believe he means the division of two complex numbers lead to the substraction of their arguments. But this was by no means pulled out of a hat. It is a principle covered in one of the previous lectures. Thank you very much for the videos Professor Gross, And i find it amazing that after all these years you are still in contact and commenting on the youtube video.
Thank you very much sir for demystifying complex variables, because in all my life, complex variables seemed to be weird and made no sense in the practical world to me. Now i can see how powerful they are all thanks to your amazing explanation sir. Looking forward for more such lectures from MIT.
Robert, RU-vid has only the video portion of the course. In the written material we point out that Part 3 of “Calculus Revisited” is meant to give an overview of three of the courses that usually follow the calculus sequence (Parts 1 and 2). In that context it was not mean to include everything that a formal course would contain. In summary, the purpose of Part 3 was to give the students a “leg up” if and when they enrolled in a complex variables course
Hi, wanna learn more about the application of conformal mappings in aerodynamics? Check this app out: play.google.com/store/apps/details?id=com.pand3.studios.engineering.joukowskisimul Thank you.
I love these vids. The most intellectual movies are in black and white anyway. I just get comfortable, break out the popcorn and Iook forward to some more 'inescapable conclusions' in complex analysis. It's like the Citizen Kane of math tutorials.
After more than 40 years I might not remember but if you could be a bit more specific I will try to answer your question. I hope you are finding the course to be useful to you.
The teaching is very incremental and I enjoy the lecture a lot. IMHO, this is THE way Complex Analysis should be taught to engineers. By the way this is the first time I see a black chalk :D
Dear Mr. Herbert Gross, I watch your lectures quite often ,and I have some confusion about conformal mapping. I would be even more grateful and appreciative if you cleared. I am aware certain real life problem such as heat distribution over irregular shapes can be reduced to simpler shape, solved and inverted again. However, I am not convinced and find it hard to accept the legitimacy and validity of this operation (mapping). In your lecture, you have compared this to change of variables in calculus. While this comparison is useful, I still find it difficult to accept the legitimacy of the operation. Is the validity guaranteed because Laplace equation is preserved after conformal mapping? Or is the validity guaranteed because the neighborhood remains the same before the transformation and after the transformation (no new point creeps into the neighborhood of another point for which it wasn't a neighbor before the transformation). I don't get the importance/significance of angles being preserved after the transformation. What are the implications that we can profit from? I have tried to justify conformal mapping to myself by imagining that I am wearing a spectacles that elongates twice the width and height of what I see through the spectacles? I kind of get the effects would be unchanged if I wear the specs or don't wear the specs. FOR EXAMPLE, if everything what I see(all objects of earth) was made twice in length and width, the level of difficulty for getting into a bus would remain the same. But, if only the bus were transformed twice its size, then it would probably become more difficult. After all , I am modifying only my perception of the objects. But, then I get confused with this spectacles analogy too. What if there is a heat source? Should I make the energy or temperature of heat source double so that effects on the objects nearby remains the same? I would appreciate if you aided me in getting rid of my confusion. I am just confused on the validity of conformal mapping. Do you think if I watch the lecture again my confusion would be cleared? Thank you anyway for your lectures. I enjoyed the one on Laplace transform.
Alas! I’m sorry that the 88 year old Herb Gross has not retained as much knowledge of complex variables as the 42 year old Herb Gross in the video. More seriously, except for the 5 years I spent at MIT producing the “Calculus Revisited” video trilogy, the rest of my career (55 years) was spent at the community college teaching arithmetic and basic algebra to mathematically at-risk adult learners. So the passage of time has basically pushed the fine points of complex analysis out of my memory banks. So I think it’s best if I don’t take the risk of trying to give you a meaningful answer to your question. I am hoping that a viewer who sees your message will give you a decent answer. It is possible that viewing the video again will help and I’d like to believe that Google can supply you with many resources that might be helpful. I surfed the Internet and found many references such as www.iiserpune.ac.in/~p.subramanian/conformal_mapping1.pdf. Perhaps one of the references will address your question directly. I wish I could have been more helpful.
Thank you for your kind words. I am often perceived as “Mister Nice Guy” but in your case it is less of being nice but rather more of being respectful to you. I feel that anyone who is taking the time to write to me (or asks a question via RU-vid), deserves a thoughtful and honest response. It also reminds me to remember that sometimes the most authentic answer to a question is “I don’t know”. Besides, at age 89 what else do I have to do? :)
Hi, wanna learn more about the application of conformal mappings in aerodynamics? Check this app out: play.google.com/store/apps/details?id=com.pand3.studios.engineering.joukowskisimul Thank you.
Thanks, Joseph. Your comment highlights the value of a video versus a live lecture. Namely, using the rewind, pause and fast forward keys, you, not the lecturer, control the pace of the lecture. It reminds me of the joke person being so “dumb” that it took him 2 hours to watch “60 Minutes”. Ironically, that defines the power of the video. You can take as long as you want to internalize its content! I hope you continue to work at your own pace and enjoy the rest of my videos!
Great lecture, as always! But here's something that isn't very clear to me. On the solution of the first exercise of this chapter (1.6.1), on part a) it says that if we map a "sufficiently small" area R centered at z_0, under a function f, then the size of the new area is the size of R multiplied by the magnitude of f' at z = z_0. But, correct me if I'm wrong, this wasn't mentioned in the lecture. How do we arrive at that conclusion?
RU-vid has the videos but not the written materisls (which are on the OCW website). My recollection is that what I refer to as "Blocks" in the written material is now referred to as "Parts" on the MIT website.
What is really interesting to me in terms of some sort of time warp is that the Herb Gross in the videos is, and always will remain, 42 years old. And hopefully he will continue to teach others even when this 89 year old Herb Gross is no longer here. Such is the power of the Internet!
Hi again Robert. There is a lot of subjectivity in deciding what to include and what to omit in an overview. So I might have omitted some topics you would have included (and included some that you would have omitted). However to the best of my ability, I never pulled a rabbit out of the hat.
Herb Gross has published three "Calculus Revisted" courses on MIT OpenCourseWare ("Single Variable Calculus" ocw.mit.edu/RES18-006F10, "Multivariable Calculus" ocw.mit.edu/RES18-007F11, "Complex Variables, Differential Equations, and Linear Algebra" ocw.mit.edu/RES18-008F11 ) and has a personal website ( www.mathasasecondlanguage.org ) with even more materials on arithmetic and algebra. Best wishes on your studies!
Hi Robert, I have no idea what you are referring to. I never discussed subtraction using polar coordinates, so what did I pull out of the hat? And while it isn’t mandatory, I would appreciate it if you would try to come across a little less authoritatively in your comments. Otherwise it is difficult for me to have a reasonable discussion with you.
Actually I was born and raised in Boston. My accent may be due in part to immigrant parents. My experience has been that people from New York tell me I have a Boston accent and people from Boston tell me I have a New York accent. However until my “Calculus Revisited” course was uploaded no one ever mentioned that I sound like Christopher Walken. Since then, many viewers have made that comment. Go figure…:)
Thank you. However, mapping transformations from R planes to the C plane should be inclusive to (x,y,z) and (I,j,k) variables at all times. The objective this mapping is to cease the margin of error that is generated and requires less error points. After watching this, it has been brought to my attention to ask the following: Application- volume area (V) requires a thickness point. Why can’t we map this through the complex field and only through the (u,v) ~ height to length vector? I believe that mapping should be inclusive to an imaginary representation language that can also define z and the point (k)… The proof is extremely hard to compute but you never know one day! 🎉
I completely lost track at the determinant part (I know what they are, but I don't get the application in this case) Can someone tell me what I should read or understand before continuing this course?
which minute is it? Take a look at this for a function to be locally invertible determinant of it's jacobian must not be zero . en.wikipedia.org/wiki/Inverse_function_theorem
At 1:25 he starts to recall some part of the course (which he calls "block 4 or block 3"), and I have no idea what he is referring to. Can anyone help?
I am not sure what your question is asking. I have no recollection concerning a review lesson. Can you give me a better description of what you are looking for and I will try my best to accommodate you.
well I was trying to figure out what lecture talked about the determinant of the jacobian matrix when talking about mappings but I just found your series of lectures on multivariable calculus which answers my question so thank you very much. By the way I just started to watch your lectures and I love them! Thank you for having done such a great job
Thanks Xavier. I am very pleased that my videos, which were produced 45 years ago, are still helping viewers and it is my hope that they will continue to do so even when I am no longer here! Developing tis course was a lot of hard work but at the same it was a labor of love.
Thanks imoc. Your use of the word “lost” reminded me of an story about optimism. An old man is walking down the street wearing only one shoe. A passerby notices this and says, “Oh you poor man; I see that you have lost a shoe!. “Nope”, the old man replied with a huge grin, “I found one!” This has absolutely nothing to do with your comment but I just felt like sharing it with you.
